The research concerns the transport of solute materials through the vascular bed of the heart and the development of quantitative kinetic expressions describing the rapidity of movement of ionic solutes such as sodium, potassium, and calcium and of metabolic substrates such as sugars, amino acids and fatty acids from the blood into the myocardial cells where they are used in generating energy for contraction. The investigations provide mathematical descriptions of the transport of material which does not leave the blood stream such as red cells or albumin, but does become dispersed in its transport through the organ. The next stages have required detailed studies on the anatomy of the capillary bed and of the permeability of the capillary membrane to solute molecules. Additional studies concern the rate of diffusion of tracer solutes through the extracellular extravascular space which these solutes traverse in going from the blood into the cell. The cell membrane is the next barrier. For many solutes, transport across the cell membrane is facilitated by carrier proteins, some of which act as actively transporting pumps requiring energy consumption, and for which precise descriptions are now needed. An example of an active pump is the sodium pump, which may be inhibited by digitalis. A secondary carrier mechanism, utilizing the driving force produced by the sodium pump, is a sodium-calcium exchange carrier via which the intracellular calcium concentration is kept very low. The strength of contraction in normal hearts and in heart failure may be understood quantitatively when accurate descriptions on these solute transport mechanisms are worked out. Presently work is proceeding in parallel on each of these types of exchanges, with the objective of integrating the kinetic expressions into an overall description of heart muscle contraction.